Connections in Tangent Categories

TL;DR

This paper explores the concept of connections within tangent categories, generalizing key differential geometry results like Bianchi identities and parallel transport to an abstract categorical framework.

Contribution

It introduces a categorical framework for connections, deriving classical differential geometry identities and structures in an abstract setting.

Findings

Derivation of Bianchi identities in tangent categories

Formulation of curvature and torsion identities abstractly

Establishment of parallel transport in categorical context

Abstract

Connections are an important tool of differential geometry. This paper investigates their definition and structure in the abstract setting of tangent categories. At this level of abstraction we derive several classically important results about connections, including the Bianchi identities, identities for curvature and torsion, almost complex structure, and parallel transport.